Key Concepts and Formulas

• General Equation of a Circle: $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. Also, $x^2 + y^2 + 2gx + 2fy + c = 0$ represents a circle with center $(-g, -f)$ and radius $\sqrt{g^2 + f^2 - c}$.
• Circle Touching the x-axis: If a circle touches the x-axis, the distance from the center to the x-axis equals the radius, i.e., $|k| = r$.
• Intercept on the y-axis: The length of the intercept made by the circle on the y-axis is $2\sqrt{r^2 - h^2}$, where $(h,k)$ is the center and $r$ is the radius.

Step-by-Step Solution

Step 1: Determine the center and radius from the tangency condition.

The circle touches the x-axis at $(a, 0)$ and lies below the x-axis. This implies the center has coordinates $(a, -r)$, where $r$ is the radius and $a > 0$. The y-coordinate is negative because the circle lies below the x-axis. Therefore, $h = a$ and $k = -r$.

Step 2: Formulate the equation of the circle.

Substituting the center $(a, -r)$ into the standard equation of a circle, we have: $(x - a)^2 + (y + r)^2 = r^2$ Expanding the equation: $x^2 - 2ax + a^2 + y^2 + 2ry + r^2 = r^2$ Simplifying, we get: $x^2 + y^2 - 2ax + 2ry + a^2 = 0 \quad (*)$

Step 3: Relate the radius to the y-intercept length.

The circle cuts off an intercept of length $b$ on the y-axis. Using the formula for the y-intercept, we have: $b = 2\sqrt{r^2 - a^2}$ Squaring both sides: $b^2 = 4(r^2 - a^2)$ Rearranging to express $r^2$: $b^2 = 4r^2 - 4a^2$ $4r^2 = b^2 + 4a^2$ $r^2 = \frac{b^2}{4} + a^2 \quad (**)$

Step 4: Compare with the given general equation.

The given equation is $x^2 + y^2 - \alpha x + \beta y + \gamma = 0$. Comparing this with equation $(*)$, we have:

• $-\alpha = -2a \implies \alpha = 2a$
• $\beta = 2r$
• $\gamma = a^2$

Step 5: Express the ordered pair $(2a, b^2)$ in terms of $\alpha, \beta, \gamma$.

We want to find $(2a, b^2)$ in terms of $\alpha, \beta, \gamma$.

• From $\alpha = 2a$, we have $2a = \alpha$.

• From $\beta = 2r$, we have $\beta^2 = 4r^2$. From equation $(**)$, we have $b^2 = 4r^2 - 4a^2$. Substituting $\beta^2 = 4r^2$ and $\gamma = a^2$, we get: $b^2 = \beta^2 - 4\gamma$

Therefore, the ordered pair $(2a, b^2)$ is $(\alpha, \beta^2 - 4\gamma)$.

Common Mistakes & Tips

• Carefully distinguish between the coefficients $\alpha$, $\beta$, $\gamma$ in the given equation and the parameters $a$, $b$, $r$ that define the geometry of the problem.
• Remember that since the circle is below the x-axis, the y-coordinate of the center is negative.
• Ensure you use the correct formula for the length of the y-intercept.

Summary

We derived the equation of the circle based on the tangency condition and the y-intercept length. By comparing the derived equation with the given general equation, we related the coefficients $\alpha, \beta, \gamma$ to the parameters $a, b, r$. Finally, we expressed the ordered pair $(2a, b^2)$ in terms of $\alpha, \beta, \gamma$, obtaining $(\alpha, \beta^2 - 4\gamma)$.

Final Answer

The final answer is \boxed{(\alpha, \beta^2 - 4\gamma)}, which corresponds to option (B).